 6th lecture of the course called game theory and economics. Before we start, let me recapitulate what we have done in the previous lecture. We have been discussing the model of Barthra oligopoly and we have been discussing the application of Nash equilibrium in find out what will be the equilibrium in such a game, in such a market and we have seen that the equilibrium in the Barthra model will be such that all the firms will, if there are two firms, all the firms will charge the same price in the equilibrium and they will charge, they will earn zero profits. If the number of firms goes up, if there are more than two firms, then also all the firms which are there in the market will earn zero profit. So, Barthra model is a particular model where if the cost of production of all the firms is equal, there is no difference in their technology and their cost of production, then there is no reason why any firm will get any positive profit. All the firms will be on equal footing and they will earn the same zero profit. So, that is what we have seen in the previous lectures. We have already seen other aspects of the Barthra model, for example, if suppose there are two firms, but their cost of production differs. If their cost of production differs, then there is, we have an asymmetry. If there is asymmetry, then we have seen that under certain conditions, the firm which is having a lower cost of production will earn positive profit in the equilibrium. So, that is there. So, this is like the Cournot outcome. If you remember in Cournot also, if the firms are having different cost of production and then the firm which is having a lower cost of production will have an upper hand in the sense that it will produce more output, it will earn higher profit and eventually it may happen that the other firm is not producing anything. Here also, what will happen is that if my cost of production is less than my rival, then I will cater to the market entirely and my rival will not produce any output. It will not be able to sell anything in the market. Not only that, in this case, I will be earning some positive profit, which was not there if the cost of production of the two firms where was equal. So, this is what we have done, we have seen. Now, let us do another small exercise to look at another aspect of Barthra model in particular. What will happen if there is a fixed cost of production? So, this is the setting. Consider Barthra's game in which the cost function of each firm i is given by c i q i is equal to f plus c q i for q i greater than 0 and c i 0 is equal to 0, where f is positive and less than the maximum of p minus c multiplied by alpha minus p with respect to p. The price p that satisfies p i p minus c multiplied by alpha minus p is equal to f and is less than the maximizer of p minus c multiplied by alpha minus p. Show that if firm i gets all the demand, when both firms charge the same price, then p 1 p 1 is the Nash equilibrium. Show also that no other pair of prices is a Nash equilibrium. So, this is the question. In terms of, if I have to show this in terms of diagram, how will it look like? What is happening is that there are two firms, 1 and 2 and their cost of production is the following where f is positive. One thing to notice is that we are back to the old framework where the unit cost of production of both the firms is equal. It is given by small c. So, the cost of production unit cost of production is not differing. What is important is that here I have a fixed cost which is given by f and we have already seen that if my level of output is 0, if I am not producing anything, then I do not have to bear that fixed cost. That is important because generally in economics it is assumed that if I am not producing anything, then also I have to bear this fixed cost f because f is not variant with respect to q i. So, even if q i is 0, f is there, but here we are assuming that if q i is 0, then f also it just vanishes and if f is positive, if q i is positive, only then we have this function. So, that is the setting. We are also informed that p i is that price which solves p minus c alpha minus p is equal to f p 1. This is the price p 1 which solves this equation and this price we are also told is less than the price which maximizes. So, suppose p m maximizes p minus c alpha minus p, then p 1 is less than p n. That information we have. What we need to show is that when both the firms charge the same price p 1, p 1, then that is a Nash equilibrium. So, to show and we are given the assumption that if both the firms charge the same price, then firm 1 gets the demand, all the demand. So, if both the firms are charging the same price, firm 2 is not getting any part of the demand. It is not getting any profit therefore. So, in terms of diagram, this is the old Barthra diagram. So, this is c, this is suppose alpha and this is the profit function. At this level p m profit is getting maximized. So, at this level pi is highest. Suppose alpha is f is given by this value and what is the significance of f is that it tells us what is that level of price at which the profit earned by a firm is just equal to 0. So, if a firm is a monopolist, suppose there is no rival firm and it charges a price p 1, then its profit is going to be just equal to 0. Why it is so? Because of this, because what is profit? If I consider a monopolist, what is profit after all? It is p q minus c i q i p q i minus c i q i minus f. This is the total revenue and this is the total cost. So, from here I can take q i common. So, this is just p minus c minus f because c i q i is equal to c q i and what is q i? q i is nothing but what is the demand? Demand in the market is alpha minus p p minus c minus f. So, this is the profit under this sort of assumption where there is a fixed cost and if there is no rival, I am getting the entire market. This is the profit. So, in this case, if I charge a price p 1, then this entire thing becomes 0 by this because of this fact. Therefore, p 1 is given by this point on the horizontal axis. What we need to prove is that p 1 p 1 is a Nash equilibrium. What is the proof? One can take this as a homework and try to see that suppose the rule of dividing the market was same as it was before. For example, if the prices are equal, the market is split equally between the two firms. So, if prices are equal, market is divided equally, then is there any Nash equilibrium? One can show it that if the rule is that if the two firms are charging the same price, then the market is divided equally, then there can be no Nash equilibrium. In this case, in the exercise, we are having a Nash equilibrium because when the prices are equal, firm 1 is getting the entire market. So, that is more or less what we had to discuss about Barthra model. Let me take up the next topic, which is the case of electoral competition. So, electoral competition by this, what we mean is that we are trying to look at how people vote and how number of candidates, when the people are voting, number of candidates are decided. More importantly, what are the agenda said by the candidates? If they want to win the election or if they have some ideological persuasions, so what will be the agenda that will be said by the candidates? Who will win in a particular equilibrium situation? So, these are some of the points and if suppose there are some costs involved in standing in the election, if you want to be a candidate, you have to bear some cost, then does it hamper the election process? Does the number of people who are running for the election, does that go down? So, there are these important issues, which we want to address in a very elementary manner in this section of electoral competition. So, the framework that we have is the following. There is a continuum of numbers. Each number represents the favorite position at least one voter. So, I have this real line for example. In this real line, if I pick up any point, then this point represents a number. This number will be the favorite position of at least one voter. So, we can imagine that each point in this real line is basically corresponding to at least one voter. There can be more than one voter who has the same favorite position. Now, when we say favorite position, what exactly do we mean? What we mean is that the political preferences of the voters can be represented in a unidimensional scale. So, this is a very simplifying assumption, mind you, because my political preferences might be multidimensional. It can be having 2 or 3 or more than that set of points, set of numbers and this set of numbers might be a representative of what my political preferences are. But since this is a very elementary exercise, what we are proposing is that my entire political preferences, what I like, what I dislike, can be represented by a single number. This number can be higher, this number can be lower, etcetera. However, this may seem a little outlandish to begin with. It is to be remembered that when we say, when we discuss political issues, we talk about leftist and rightist. So, at the back of our mind, we have this unidimensional scale. Some person is preferring some policy, which are to the left, which means maybe in this line, you are going to this direction. We are preferring some point here and someone is rightist, which means he is preferring some point here. So, that can be visualized in that sense. So, it is not so uncommon to visualize a straight line and each point on the straight line is representing the political proclivities of a particular voter. So, this is that every point is representing the favorite position of a voter. We can think of this number to be, suppose, defense budget. So, the amount of money that the country will spend on defense is represented by a single number. Now, if I am rightist, it is possible that I like that number to be very high. Whereas, if I am leftist, I like that number to be not as high. If I am a centrist, my preferences will be between these two numbers. So, this is just an illustration, an instance of how political preferences can be represented by a single number. So, suppose x i star is this point, this is the favorite position of individual i, voter i. Now, the point is that he likes the choice of the entire country to be x i star. But if it is not x i star, then how does he rank those positions? Suppose, there is another point x 1 and there is another point x 2. In this model, we are going to assume that he ranks the x 1 and x 2, these two numbers in the following sense that further those numbers are from his favorite position which is x i star, the less he likes those numbers. So, in this case, x 1 will be preferred to him than x 2. So, in a sense that the distance between his favorite position and any arbitrary position represents his dislike for that arbitrary position. So, I can write it like this that suppose u i x i represents how do I like the position x i? Let us not write x i, suppose x 1, otherwise it will be confusing with i, suppose x 1. So, u i x 1 is representing what is the payoff of player i from the position x 1. So, if the country is taking up this policy x 1, then how does individual i like that? This can be represented by x i star, x i star is his favorite position and let us take this distance and what we are going to do is to take a square of that. Now, if I take just the distance and the square, it will mean that more the distance is better I prefer that policy which is not in fact the case. So, it is just the other way, the more distant x 1 is from x i star my dislike for that policy goes up. So, that is why I have put this negative sign and why did I square that? Why did I square that is because suppose there are two policies x 3 and x 1 and they are equidistant from x i star. So, this part is same as this part, then our model is going to assume this setting is going to assume that my dislike for x 1 and x 2 or my liking for x 1 and x 2 is the same. So, I do not differentiate, I do not distinguish whether that position is towards my left or towards my right as long as there is equidistant from my favorite position, my liking slash disliking liking is the same. However, we can relax this assumption a little bit because one can imagine that I do not like the policies which are to the right that much as I like the policies which are towards my left. So, those extensions, those added complications can be included and we can see how they can be included in a later exercise. So, this is the setting that every voter has a favorite position and he likes the policy of the country to be closest to his favorite position. The more distant the policy of the country from his favorite position, the greater is his dislike for that policy and he does not differentiate whether that policy is towards his right or left. This is just a simplifying assumption. So, given this case, we have seen that all the voters are having some favorite positions over this line. Now, we are not going to assume whether this distribution of favorite positions is of a particular kind, it is just a continuous distribution. There is no gap between any two points in this line, that is all we are trying to say. The distribution can be of any sort that is an open ended thing. What will be important is that in this distribution, there is going to be a median of this distribution. Let us call that median to be small m. So, the definition as we have known before is that half of the voters in this country will have their favorite positions either equal to m or less than m and half of the voters will have their favorite positions either equal to m or greater than m. So, that is how the median is defined. We are going to see that this m position is going to be of major importance. Now, let us talk about the candidates. The candidates, what they are trying to do is that they are trying to win the election very obviously and suppose there is a number of candidates, there are n number of candidates and they are going to announce the positions. Suppose, x 1, x 2, x 3, these three positions they are announcing and their promise is that if they are elected to the office, they are going to implement these positions. So, if candidate 1 wins, he is going to implement x 1 and if 2 wins, he is going to implement x 2 like that. Now, after they have announced their policies, the voters vote. So, suppose x 1 is here, x 2 is here, x 3 is here. How will the voters vote? That is the point. Well, given the assumptions that we have so far, it is not very difficult to see that the voters will vote for that candidate which is closest to his favorite position, their favorite positions. So, if I am here, suppose this is x i star, then the closest candidate for me is x 1. So, I am going to vote for candidate 1 who has announced x 1 because x 2 is very far away from x i star, x 3 is even further from x i star. So, you can see x 1, the difference between x 1 and x i star is not much. So, these votes from here also all will go to candidate 1, but how much? Will this voter vote for candidate 1? The answer is obviously no, he is going to vote for candidate 2. So, if I think about it little more carefully, I can figure out that there will be one point before that point, that is to the left of that point, all the voters will vote for x 1 and after that point, the voters will vote for x 2. Similarly, there will be some point here between x 2 and x 3 such that, suppose this point is p and this point is q. So, before q and after p, the voters will vote for x 2 and after q, every voter will vote for candidate 3. That we can figure out because all these points are closer to x 3 than to x 2 and it is not very difficult to figure out that p is nothing but x 1 plus x 2 divided by 2. So, p is dividing this distance between x 1 and x 2. If you are to the right of p, you are closer to x 2 than to x 1. If you are to the left of p, you are closer to x 1 than to x 2, that is why p is dividing this line of x 1 and x 2 and similarly, q is dividing x 2 and x 3. So, all these voters here are going to vote for 1, voters here vote for 2. So, this is how the voting takes place and the candidates know this and since the candidates know this, what they want to do is that they want to win the election and so, they want to garner, they want to get as many votes as possible. In this context, it is important to note that there are no ideological persuasions. We are going to assume that the candidates are not concerned about the positions that they take. Only thing that they are interested in is to win the election. We can again relax this assumption a little bit, but this is a simplifying assumption to begin with. We are going to assume that candidates do not bother about the positions that they take as long as they win, that is the best thing that can happen to them. Now this is the setting then. In this case, if we have this setting, then the questions that we want to ask are how the candidates will choose their positions that they are announcing. For example, how x 1, x 2, x 3 are decided and if they have decided they are announced their positions, then how the voters are going to vote and who will win and if at all people will win or will there be a tie. This is the question that we are interested in. Remember, we are talking in terms of game theory, we are in a game theoretic framework. It is better to specify this setting in terms of the language of game theory. Players, the candidates, n in number, this is important. This is important because we are going to assume that the voters are not playing the game. The voters are going to vote in a non-strategic fashion. So, they are not going to calculate this other voter is voting for him. So, I should vote for this candidate so that my favorite candidate wins nothing like that. The voters are just looking at the announcements made by the candidates who are running the election and they are choosing that candidate who is closest to their favorite position. That is all. It is the candidates who are trying to act strategically and deciding what announcements to make so that they can win. What are the actions that they are taking? The actions are basically numbers. My actions is set of positions and these positions are represented by numbers. So, the actions that I take, I can take are basically some numbers. I can choose any of those numbers and that will be my action. Another thing I did not mention is that if there are some voters on this line P, P is the middle point between X1 and X2, then these voters are going to be equally divided between candidate 1 and candidate 2 because they are on the borderline. Preferences are very intuitive and obvious. The candidates want to win. So, winning is the best possible thing that they can do. So, winning is best by n, suppose. So, if I win, I get n. What is the second best? Second best is that I do not win outrightly, but I tie with some other candidate in the first place. So, I am in the first place, that is true, but this is not an outright win for me. This will be represented by k if I ties with n minus k other candidates. So, this is the second case of this is the winning and this is the tying. If I tie, then I get k. k is an integer and k can vary between n minus 1 and 1. If k takes the highest value, that is n minus 1, then basically by putting k is equal to n minus 1, from here I get 1. So, this is the case when I is tying with just one other candidate in the first place. So, there are two winners here and if there are two winners, the payoff that each of them gets is given by n minus 1 and this goes on rising. This payoff goes on declining as the people who are tying in the first place goes on rising. For example, let us take the lowest possible value of k. If k is equal to 1, then I is tying with n minus 1 other candidates, basically he is tying with everyone. Nobody is a winner and nobody is a loser either. Everybody is tying in that case, he is getting a payoff of just 1 and lastly, 0 if I loses. So, this is the worst possible situation that you contest a election and there is at least one candidate who has got more votes than you and therefore, you lose and therefore, you get 0. So, this is the set of preferences. Now, what will be the Nash equilibrium? What we are going to do is that like before in the while trying to find out the Nash equilibrium in other problems, we are going to find out what are the best response functions of candidates and try to see at what point they intersect with each other. But to make it tractable, we are going to assume that n is equal to 2. So, there are just two candidates who are competing with each other and trying to win the election. Now, let us look at this problem from the point of view of player 1 that is candidate 1. Now, depending on the position announced by candidate 2, his best response will be different and this announcement by candidate 2 can take different kinds of values. What is important is that that value is it greater than or less than m or is it just equal to m? That is an important question to ask. So, x 2 is the announcement made by candidate 2, x 2 can be of different values, it can be less than m, it can be more than m. If it is less than m, suppose announcement by 2 x 2 is less than m, then what is best for candidate 1 to do? Candidate 1 will obviously in this case announce something more than x 2 because if it announces something less than x 2, then it is getting these words but all these words are going to x 2 then, which is not a good thing to do because I know that m is here, so half of the voters are to the right of m, these words person 2 will get, these words person 2 will get, so he is going to win. So, person 1 that is candidate 1 is never going to announce something less than x 2. Will he announce equal to x 2? If he announces equal to x 2, there is going to be a tie because in that case all the voters are going to be divided between the two candidates, so that is going to be a tie. The thing to do is that for candidate 1 that he will announce something more than x 2, so if that is the first conclusion that we can draw from here but is it an arbitrary value as long as it is greater than x 2? The answer is no because suppose here is m and here is x 2, if one announces something here greater than m and too much greater than m, x 1 is too much far away from m, then all these votes will come to candidate 2 and there will be some votes from here also which will come to candidate 2 because for this voter this distance is less than this distance, so he will vote for candidate 2 and these voters are in addition to the half of the voters that candidate 2 is already getting, so in that case candidate 2 will win. So, for candidate 1 to win he will announce something more than x 2 that is true but at the same time that more should not be too much, he will keep himself close to x 2 and not go too much further away from x 2. So, if I have to draw this in a more clearer diagram, if m is here x 2 is here then x 1 should be here such that x 1 plus x 2 divided by 2 is less than m, if it is less than m then candidate 1 is getting all these votes which is greater than half, if it is equal to m then again the voters are going to be equally divided. So, one thing is that x 1 is greater than x 2 and second thing is that x 1 plus x 2 divided by 2 should be less than m and if I simplify this x 1 plus x 2 less than 2 m or x 1 is less than minus x 2. So, in short if x 2 is less than m x 1 which is equal to b 1 x 2 should be is given by x 1 such that x 1 is greater than x 2 but less than twice m minus. So, this is one best response function we are going to construct other best response functions in the next lecture and try to find out what will be the equilibrium in this case. To recapitulate what we have done in this lecture is that we have finished our discussion of Valtran oligopoly and we have started discussing the electoral competition games and we have trying to find out what are the best response functions. Thank you. There is a fixed cost of production with cost function given by c i q i is equal to f plus c q i for q i greater than 0 and cost is equal to 0 for q i is equal to 0 profit maximizing price is higher than p star what is p star? p star is the price where the profit is 0 and market is divided equally between firms if the prices are same is there an equilibrium. Let us try to visualize this what is happening here. So, what I am drawing is the profit function of any firm and this is the price that is charged by that firm this is the price at which profit is maximized. So, let us call it p m and suppose this is the p star price and this is the value of f the fixed cost and this point of intersection is the point of c and this is the point alpha. Now, is there an equilibrium and our claim is that there is no equilibrium here what could have been the probable candidate for equilibrium for example, p star p star could have been a probable candidate for equilibrium because apparent reason is we know that f is equal to alpha minus p star p star minus c at p star the total profit is 0. So, this must hold the fixed cost is equal to the profit from the variable cost component, but here at p star market is equally divided. So, profit at p star is we can calculate this to be alpha this which is negative. So, rather than earning a negative profit a firm will deviate and charge something more and earn 0 profit. So, p star p star not equilibrium if we take any other pair of prices higher than p star then some firm is earning a positive profit if the prices are different the lower price charging firm is earning some positive profit in that case other firm will undercut that firm. So, that cannot be an equilibrium another consideration we could take is this price pair where this holds. So, what is happening is that at p bar if both the firms charge p bar let us suppose this is p bar then their individual profit is 0, but this is not an equilibrium for the reason that each firm will then will have a tendency to undercut the other firm and earn a positive profit you know at this any price less than p bar you are getting the enter market or the sharing the market and your profit will be positive. So, by this logic there is no equilibrium no equilibrium briefly introduce the model of the liberal competition of hoteling. So, what we have a continuum of voters favorite positions and n candidates players they are the players of the game contest the election in the election each candidate announces a number representing the policy that you will implement and depending on the announcement of the candidates the voters vote the closer the position is to your favorite position the better of you are if the candidates announcement is something far away from your favorite position you are less likely to vote for that candidate. And what are the preferences of the candidates? Candidates want to win outright this is the first if not tying is preferable to losing tying in the first place is preferable to losing in tying tying with fewer candidates is preferable. So, this is the game and this is the basic setting of the game. Thank you.